Answers

If you have 80% moss, probably easier to start again, but change the conditions or you will end up with the same again even if you re-turf/seed.
Try increasing light - is there a tree overhanging that you could raise the canopy? (chop lower branches to let more light in)
What soil are you on? compacted clay will usually give poor grass and good moss.
If you want to persevere with what you have, try scarifiing (raking all the moss out and making it look lots worse), then a hollow tyne aerator + sharp sand brushed into the holes perhaps with some seed suited to the position ( ask your local garden centre there are seed for shady sites).
Good soil + light will encourage grass. Compacted soil, wet + shade will encourage moss. Chemical treatment will not help longterm as nature will re-establish itself unless you change the basic conditions.